How do I produce an uneven offset /solidify modify on a cube?
If you are solidifying through the Solidify modifier you can use a vertex group to control thickness.

Create a vertex group and add all desired vertex to it, set solidify thickness to a known value so multiplying by a factor is easy, like say $1.0$ units, or $0.1$ Units.

This value will be multiplied by the Vertex Weight set for that vertex. Now adjust thickness on a per vertex basis adjusting vertex weight.

With with weight paint, Vertex Weight Proximity modifier, or manually.